Introduction To Binary Trees

A binary tree is made of nodes, where each node contains a "left" pointer, a "right"
pointer, and a data element. The "root" pointer points to the topmost node in the
tree. The left and right pointers recursively point to smaller "subtrees" on either
side. A null pointer represents a binary tree with no elements -- the empty tree.
The formal recursive definition is: a binary tree is either empty (represented
by a null pointer), or is made of a single node, where the left and right pointers
(recursive definition ahead) each point to a binary tree.

A "binary search tree" (BST) or "ordered binary tree" is a type of binary tree where
the nodes are arranged in order: for each node, all elements in its left subtree
are less-or-equal to the node (<=), and all the elements in its right subtree
are greater than the node (>). The tree shown above is a binary search tree -- the
"root" node is a 5, and its left subtree nodes (1, 3, 4) are <= 5, and its right
subtree nodes (6, 9) are > 5. Recursively, each of the subtrees must also obey the
binary search tree constraint: in the (1, 3, 4) subtree, the 3 is the root, the
1 <= 3 and 4 > 3. Watch out for the exact wording in the problems -- a "binary
search tree" is different from a "binary tree".
The nodes at the bottom edge of the tree have empty subtrees and are called "leaf"
nodes (1, 4, 6) while the others are "internal" nodes (3, 5, 9).

Binary Search Tree Niche

Basically, binary search trees are fast at insert and lookup. The next section presents
the code for these two algorithms. On average, a binary search tree algorithm can
locate a node in an N node tree in order lg(N) time (log base 2). Therefore, binary
search trees are good for "dictionary" problems where the code inserts and looks
up information indexed by some key. The lg(N) behavior is the average case -- it's
possible for a particular tree to be much slower depending on its shape.

Strategy

Some of the problems in this article use plain binary trees, and some use binary
search trees. In any case, the problems concentrate on the combination of pointers
and recursion. (See the articles linked above for pointer articles that do not emphasize
recursion.)
For each problem, there are two things to understand...

The node/pointer structure that makes up the tree and the code that manipulates
it

The algorithm, typically recursive, that iterates over the tree

When thinking about a binary tree problem, it's often a good idea to draw a few
little trees to think about the various cases.

Typical Binary Tree Code in C/C++

As an introduction, we'll look at the code for the two most basic binary search
tree operations -- lookup() and insert(). The code here works for C or C++. Java
programers can read the discussion here, and then look at the Java versions in Section 4.
In C or C++, the binary tree is built with a node type like this...

struct node {
int data;
struct node* left;
struct node* right;
};

Lookup()

Given a binary search tree and a "target" value, search the tree to see if it contains
the target. The basic pattern of the lookup() code occurs in many recursive tree
algorithms: deal with the base case where the tree is empty, deal with the current
node, and then use recursion to deal with the subtrees. If the tree is a binary
search tree, there is often some sort of less-than test on the node to decide if
the recursion should go left or right.
/*

/*Given a binary tree, return true if a node
with the target data is found in the tree. Recurs
down the tree, chooses the left or right
branch by comparing the target to each node.*/
static int lookup(struct node* pNode, int target)
{
// 1. Base case == empty tree
// in that case, the target is not found so return false
if (pNode == NULL) {
return(false);
}
else {
// 2. see if found here
if (target == pNode->data)
return(true);
else {
// 3. otherwise recur down the correct subtree
if (target == pNode->data)
return(lookup(pNode->left, target));
else
return(lookup(pNode->right, target));
}
}
}

The lookup() algorithm could be written as a while-loop that iterates down the tree.
Our version uses recursion to help prepare you for the problems below that require
recursion.

Pointer Changing Code

There is a common problem with pointer intensive code: what if a function needs
to change one of the pointer parameters passed to it? For example, the insert()
function below may want to change the root pointer. In C and C++, one solution uses
pointers-to-pointers (aka "reference parameters"). That's a fine technique, but
here we will use the simpler technique that a function that wishes to change a pointer
passed to it will return the new value of the pointer to the caller. The
caller is responsible for using the new value. Suppose we have a change() function
that may change the the root, then a call to change() will look like this...
// suppose the variable "root" points to the treeroot = change(root);
We take the value returned by change(), and use it as the new value for root. This
construct is a little awkward, but it avoids using reference parameters which confuse
some C and C++ programmers, and Java does not have reference parameters at all.
This allows us to focus on the recursion instead of the pointer mechanics.

Insert()

Insert() -- given a binary search tree and a number, insert a new node with the
given number into the tree in the correct place. The insert() code is similar to
lookup(), but with the complication that it modifies the tree structure. As described
above, insert() returns the new tree pointer to use to its caller. Calling insert()
with the number 5 on this tree...
2 / \ 1 10
returns the tree...
2 / \ 1 10 / 5
The solution shown here introduces a newNode() helper function that builds a single
node. The base-case/recursion structure is similar to the structure in lookup()
-- each call checks for the NULL case, looks at the node at hand, and then recurs
down the left or right subtree if needed.
/* Helper function that allocates a new node with the given data and NULL left and right pointers.*/struct node* NewNode(int data) { struct node* node = new(struct node); // "new" is like
"malloc" node->data = data; node->left = NULL; node->right = NULL; return(node);}

/* Give a binary search tree and a number, inserts a new node with the given number in the correct place in the tree. Returns the new root pointer which the caller should then use (the standard trick to avoid using reference parameters).*/struct node* insert(struct node* node, int data) { // 1. If the tree is empty, return a new, single node if (node == NULL) { return(newNode(data)); } else { // 2. Otherwise, recur down the tree if (data <= node->data) node->left = insert(node->left, data); else node->right = insert(node->right, data); return(node); // return the (unchanged) node pointer }}

The shape of a binary tree depends very much on the order that the nodes are inserted.
In particular, if the nodes are inserted in increasing order (1, 2, 3, 4), the tree
nodes just grow to the right leading to a linked list shape where all the left pointers
are NULL. A similar thing happens if the nodes are inserted in decreasing order
(4, 3, 2, 1). The linked list shape defeats the lg(N) performance. We will not address
that issue here, instead focusing on pointers and recursion.

Section 2 -- Binary Tree Problems

Here are 14 binary tree problems in increasing order of difficulty. Some of the
problems operate on binary search trees (aka "ordered binary trees") while others
work on plain binary trees with no special ordering. The next section,
Section 3, shows the solution code in C/C++. Section 4
gives the background and solution code in Java. The basic structure and recursion
of the solution code is the same in both languages -- the differences are superficial.
Reading about a data structure is a fine introduction, but at some point the only
way to learn is to actually try to solve some problems starting with a blank sheet
of paper. To get the most out of these problems, you should at least attempt to
solve them before looking at the solution. Even if your solution is not quite right,
you will be building up the right skills. With any pointer-based code, it's a good
idea to make memory drawings of a a few simple cases to see how the algorithm should
work.

1. build123()

This is a very basic problem with a little pointer manipulation. (You can skip this
problem if you are already comfortable with pointers.) Write code that builds the
following little 1-2-3 binary search tree...
2 / \ 1 3
Write the code in three different ways...

a: by calling newNode() three times, and using three pointer variables

b: by calling newNode() three times, and using only one pointer variable

c: by calling insert() three times passing it the root pointer to build up the tree

(In Java, write a build123() method that operates on the receiver to change it to
be the 1-2-3 tree with the given coding constraints. See Section 4.)
struct node* build123() {

2. size()

This problem demonstrates simple binary tree traversal. Given a binary tree, count
the number of nodes in the tree.
int size(struct node* node) {

3. maxDepth()

Given a binary tree, compute its "maxDepth" -- the number of nodes along the longest
path from the root node down to the farthest leaf node. The maxDepth of the empty
tree is 0, the maxDepth of the tree on the first page is 3.
int maxDepth(struct node* node) {

4. minValue()

Given a non-empty binary search tree (an ordered binary tree), return the minimum
data value found in that tree. Note that it is not necessary to search the entire
tree. A maxValue() function is structurally very similar to this function. This
can be solved with recursion or with a simple while loop.
int minValue(struct node* node) {

5. printTree()

Given a binary search tree (aka an "ordered binary tree"), iterate over the nodes
to print them out in increasing order. So the tree...
4 / \ 2 5 / \ 1 3
Produces the output "1 2 3 4 5". This is known as an "inorder" traversal of the
tree.
Hint: For each node, the strategy is: recur left, print the node data, recur
right.
void printTree(struct node* node) {

6. printPostorder()

Given a binary tree, print out the nodes of the tree according to a bottom-up "postorder"
traversal -- both subtrees of a node are printed out completely before the node
itself is printed, and each left subtree is printed before the right subtree. So
the tree...
4 / \ 2 5 / \ 1 3
Produces the output "1 3 2 5 4". The description is complex, but the code is simple.
This is the sort of bottom-up traversal that would be used, for example, to
evaluate an expression tree where a node is an operation like '+' and its subtrees
are, recursively, the two subexpressions for the '+'.
void printPostorder(struct node* node) {

7. hasPathSum()

We'll define a "root-to-leaf path" to be a sequence of nodes in a tree starting
with the root node and proceeding downward to a leaf (a node with no children).
We'll say that an empty tree contains no root-to-leaf paths. So for example, the
following tree has exactly four root-to-leaf paths:
5 / \ 4
8 / / \ 11 13 4 / \
\ 7 2
1
Root-to-leaf paths:
path 1: 5 4 11 7 path 2: 5 4 11 2 path 3: 5 8 13 path 4: 5 8 4 1
For this problem, we will be concerned with the sum of the values of such a path
-- for example, the sum of the values on the 5-4-11-7 path is 5 + 4 + 11 + 7 = 27.
Given a binary tree and a sum, return true if the tree has a root-to-leaf path such
that adding up all the values along the path equals the given sum. Return false
if no such path can be found. (Thanks to Owen Astrachan for suggesting this problem.)
int hasPathSum(struct node* node, int sum) {

8. printPaths()

Given a binary tree, print out all of its root-to-leaf paths as defined above. This
problem is a little harder than it looks, since the "path so far" needs to be communicated
between the recursive calls. Hint: In C, C++, and Java, probably the best
solution is to create a recursive helper function printPathsRecur(node, int path[],
int pathLen), where the path array communicates the sequence of nodes that led up
to the current call. Alternately, the problem may be solved bottom-up, with each
node returning its list of paths. This strategy works quite nicely in Lisp, since
it can exploit the built in list and mapping primitives. (Thanks to Matthias Felleisen
for suggesting this problem.)
Given a binary tree, print out all of its root-to-leaf paths, one per line.
void printPaths(struct node* node) {

9. mirror()

Change a tree so that the roles of the left and right pointers are swapped at every
node.
So the tree...
4 / \ 2 5 / \ 1 3
is changed to...
4 / \ 5 2 / \ 3 1
The solution is short, but very recursive. As it happens, this can be accomplished
without changing the root node pointer, so the return-the-new-root construct is
not necessary. Alternately, if you do not want to change the tree nodes, you may
construct and return a new mirror tree based on the original tree.
void mirror(struct node* node) {

10. doubleTree()

For each node in a binary search tree, create a new duplicate node, and insert the
duplicate as the left child of the original node. The resulting tree should still
be a binary search tree.
So the tree...
2 / \ 1 3
is changed to...
2 / \ 2 3 / / 1 3 / 1
As with the previous problem, this can be accomplished without changing the root
node pointer.
void doubleTree(struct node* node) {

11. sameTree()

Given two binary trees, return true if they are structurally identical -- they are
made of nodes with the same values arranged in the same way. (Thanks to Julie Zelenski
for suggesting this problem.)
int sameTree(struct node* a, struct node* b) {

12. countTrees()

This is not a binary tree programming problem in the ordinary sense -- it's more
of a math/combinatorics recursion problem that happens to use binary trees. (Thanks
to Jerry Cain for suggesting this problem.)
Suppose you are building an N node binary search tree with the values 1..N. How
many structurally different binary search trees are there that store those
values? Write a recursive function that, given the number of distinct values, computes
the number of structurally unique binary search trees that store those values. For
example, countTrees(4) should return 14, since there are 14 structurally unique
binary search trees that store 1, 2, 3, and 4. The base case is easy, and the recursion
is short but dense. Your code should not construct any actual trees; it's just a
counting problem.
int countTrees(int numKeys) {

Binary Search Tree Checking (for problems 13 and 14)

This background is used by the next two problems: Given a plain binary tree, examine
the tree to determine if it meets the requirement to be a binary search tree. To
be a binary search tree, for every node, all of the nodes in its left tree must
be <= the node, and all of the nodes in its right subtree must be > the node.
Consider the following four examples...
a. 5 -> TRUE / \ 2 7

b. 5 -> FALSE, because the 6 is not ok to the left of the 5 / \ 6 7

c. 5 -> TRUE / \ 2 7 / 1d. 5 -> FALSE, the 6 is ok with the 2, but the 6 is not ok with
the 5 / \ 2 7 / \ 1 6
For the first two cases, the right answer can be seen just by comparing each node
to the two nodes immediately below it. However, the fourth case shows how checking
the BST quality may depend on nodes which are several layers apart -- the 5 and
the 6 in that case.

13 isBST() -- version 1

Suppose you have helper functions minValue() and maxValue() that return the min
or max int value from a non-empty tree (see problem 3 above). Write an isBST() function
that returns true if a tree is a binary search tree and false otherwise. Use the
helper functions, and don't forget to check every node in the tree. It's ok if your
solution is not very efficient. (Thanks to Owen Astrachan for the idea of having
this problem, and comparing it to problem 14)
Returns true if a binary tree is a binary search tree.
int isBST(struct node* node) {

14. isBST() -- version 2

Version 1 above runs slowly since it traverses over some parts of the tree many
times. A better solution looks at each node only once. The trick is to write a utility
helper function isBSTRecur(struct node* node, int min, int max) that traverses down
the tree keeping track of the narrowing min and max allowed values as it goes, looking
at each node only once. The initial values for min and max should be INT_MIN and
INT_MAX -- they narrow from there.
/*
Returns true if the given tree is a binary search tree
(efficient version).
*/
int isBST2(struct node* node) {
return(isBSTRecur(node, INT_MIN, INT_MAX));
}
/*
Returns true if the given tree is a BST and its
values are >= min and <= max.
*/
int isBSTRecur(struct node* node, int min, int max) {

15. Tree-List

The Tree-List problem is one of the greatest recursive pointer problems ever devised,
and it happens to use binary trees as well. The problem requires an understanding
of binary trees, linked lists, recursion, and pointers. It's a great problem, but
it's complex.

Section 3 -- C/C++ Solutions

Make an attempt to solve each problem before looking at the solution -- it's the
best way to learn.

2. size() Solution (C/C++)

3. maxDepth() Solution (C/C++)

/* Compute the "maxDepth" of a tree -- the number of nodes along the longest path from the root node down to the farthest leaf node.*/int maxDepth(struct node* node) { if (node==NULL) { return(0); } else { // compute the depth of each subtree int lDepth = maxDepth(node->left); int rDepth = maxDepth(node->right); // use the larger one if (lDepth > rDepth) return(lDepth+1); else return(rDepth+1); }}

4. minValue() Solution (C/C++)

/* Given a non-empty binary search tree, return the minimum data value found in that tree. Note that the entire tree does not need to be searched.*/int minValue(struct node* node) { struct node* current = node; // loop down to find the leftmost leaf while (current->left != NULL) { current = current->left; } return(current->data);}

6. printPostorder() Solution (C/C++)

/* Given a binary tree, print its nodes according to the "bottom-up" postorder traversal.*/void printPostorder(struct node* node) { if (node == NULL) return; // first recur on both subtrees printTree(node->left); printTree(node->right); // then deal with the node printf("%d ", node->data);}

7. hasPathSum() Solution (C/C++)

/* Given a tree and a sum, return true if there is a path from the root down to a leaf, such that adding up all the values along the path equals the given sum. Strategy: subtract the node value from the sum when recurring down, and check to see if the sum is 0 when you run out of tree.*/int hasPathSum(struct node* node, int sum) { // return true if we run out of tree and sum==0 if (node == NULL) { return(sum == 0); } else { // otherwise check both subtrees int subSum = sum - node->data; return(hasPathSum(node->left, subSum) || hasPathSum(node->right,
subSum)); }}

8. printPaths() Solution (C/C++)

/* Given a binary tree, print out all of its root-to-leaf paths, one per line. Uses a recursive helper to do the work.*/void printPaths(struct node* node) { int path[1000]; printPathsRecur(node, path, 0);}/* Recursive helper function -- given a node, and an array containing the path from the root node up to but not including this node, print out all the root-leaf paths.*/void printPathsRecur(struct node* node, int path[], int pathLen) { if (node==NULL) return; // append this node to the path array path[pathLen] = node->data; pathLen++; // it's a leaf, so print the path that led to here if (node->left==NULL && node->right==NULL) { printArray(path, pathLen); } else { // otherwise try both subtrees printPathsRecur(node->left, path, pathLen); printPathsRecur(node->right, path, pathLen); }}// Utility that prints out an array on a line.void printArray(int ints[], int len) { int i; for (i=0; i<len; i++) { printf("%d ", ints[i]); } printf("\n");}

10. doubleTree() Solution (C/C++)

/* For each node in a binary search tree, create a new duplicate node, and insert the duplicate as the left child of the original node. The resulting tree should still be a binary search tree. So the tree... 2 / \ 1 3 Is changed to... 2 / \ 2 3 / / 1 3 / 1*/void doubleTree(struct node* node) { struct node* oldLeft; if (node==NULL) return; // do the subtrees doubleTree(node->left); doubleTree(node->right); // duplicate this node to its left oldLeft = node->left; node->left = newNode(node->data); node->left->left = oldLeft;}

12. countTrees() Solution (C/C++)

/* For the key values 1...numKeys, how many structurally unique binary search trees are possible that store those keys. Strategy: consider that each value could be the root. Recursively find the size of the left and right subtrees.*/int countTrees(int numKeys) { if (numKeys <=1) { return(1); } else { // there will be one value at the root, with whatever remains // on the left and right each forming their own subtrees. // Iterate through all the values that could be the root... int sum = 0; int left, right, root; for (root=1; root<=numKeys; root++) { left = countTrees(root - 1); right = countTrees(numKeys - root); // number of possible trees with this root == left*right sum += left*right; } return(sum); }}

13. isBST1() Solution (C/C++)

/* Returns true if a binary tree is a binary search tree.*/int isBST(struct node* node) { if (node==NULL) return(true); // false if the max of the left is > than us // (bug -- an earlier version had min/max backwards here) if (node->left!=NULL && maxValue(node->left) > node->data) return(false); // false if the min of the right is <= than us if (node->right!=NULL && minValue(node->right) <= node->data) return(false); // false if, recursively, the left or right is not a BST if (!isBST(node->left) || !isBST(node->right)) return(false); // passing all that, it's a BST return(true);}

15. TreeList Solution (C/C++)

Section 4 -- Java Binary Trees and Solutions

In Java, the key points in the recursion are exactly the same as in C or C++. In
fact, I created the Java solutions by just copying the C solutions, and then making
the syntactic changes. The recursion is the same, however the outer structure is
slightly different.
In Java, we will have a BinaryTree object that contains a single root pointer. The
root pointer points to an internal Node class that behaves just like the node struct
in the C/C++ version. The Node class is private -- it is used only for internal
storage inside the BinaryTree and is not exposed to clients. With this OOP structure,
almost every operation has two methods: a one-line method on the BinaryTree that
starts the computation, and a recursive method that works on the Node objects. For
the lookup() operation, there is a BinaryTree.lookup() method that the client uses
to start a lookup operation. Internal to the BinaryTree class, there is a private
recursive lookup(Node) method that implements the recursion down the Node structure.
This second, private recursive method is basically the same as the recursive C/C++
functions above -- it takes a Node argument and uses recursion to iterate over the
pointer structure.

Java Binary Tree Structure

To get started, here are the basic definitions for the Java BinaryTree class, and
the lookup() and insert() methods as examples...
// BinaryTree.javapublic class BinaryTree { // Root node pointer. Will be null for an empty tree. private Node root;

/* --Node-- The binary tree is built using this nested node class. Each node stores one data element, and has left and right sub-tree pointer which may be null. The node is a "dumb" nested class -- we just use it for storage; it does not have any methods. */ private static class Node { Node left; Node right; int data; Node(int newData) { left = null; right = null; data = newData; } } /** Creates an empty binary tree -- a null root pointer. */ public void BinaryTree() { root = null; }

/** Returns true if the given target is in the binary tree. Uses a recursive helper. */ public boolean lookup(int data) { return(lookup(root, data)); }

/** Recursive insert -- given a node pointer, recur down and insert the given data into the tree. Returns the new node pointer (the standard way to communicate a changed pointer back to the caller). */ private Node insert(Node node, int data) { if (node==null) { node = new Node(data); } else { if (data <= node.data) { node.left = insert(node.left, data); } else { node.right = insert(node.right, data); } } return(node); // in any case, return the new pointer to the caller }

OOP Style vs. Recursive Style

From the client point of view, the BinaryTree class demonstrates good OOP style
-- it encapsulates the binary tree state, and the client sends messages like lookup()
and insert() to operate on that state. Internally, the Node class and the recursive
methods do not demonstrate OOP style. The recursive methods like insert(Node)
and lookup (Node, int) basically look like recursive functions in any language.
In particular, they do not operate against a "receiver" in any special way. Instead,
the recursive methods operate on the arguments that are passed in which is the classical
way to write recursion. My sense is that the OOP style and the recursive style do
not be combined nicely for binary trees, so I have left them separate. Merging the
two styles would be especially awkward for the "empty" tree (null) case, since you
can't send a message to the null pointer. It's possible to get around that by having
a special object to represent the null tree, but that seems like a distraction to
me. I prefer to keep the recursive methods simple, and use different examples to
teach OOP.

Java Solutions

Here are the Java solutions to the 14 binary tree problems. Most of the solutions
use two methods:a one-line OOP method that starts the computation, and a recursive
method that does the real operation. Make an attempt to solve each problem before
looking at the solution -- it's the best way to learn.

7. hasPathSum() Solution (Java)

/** Given a tree and a sum, returns true if there is a path from the root down to a leaf, such that adding up all the values along the path equals the given sum. Strategy: subtract the node value from the sum when recurring down, and check to see if the sum is 0 when you run out of tree.*/public boolean hasPathSum(int sum) { return( hasPathSum(root, sum) );}boolean hasPathSum(Node node, int sum) { // return true if we run out of tree and sum==0 if (node == null) { return(sum == 0); } else { // otherwise check both subtrees int subSum = sum - node.data; return(hasPathSum(node.left, subSum) || hasPathSum(node.right,
subSum)); }}

8. printPaths() Solution (Java)

/** Given a binary tree, prints out all of its root-to-leaf paths, one per line. Uses a recursive helper to do the work.*/public void printPaths() { int[] path = new int[1000]; printPaths(root, path, 0);}/** Recursive printPaths helper -- given a node, and an array containing the path from the root node up to but not including this node, prints out all the root-leaf paths.*/private void printPaths(Node node, int[] path, int pathLen) { if (node==null) return; // append this node to the path array path[pathLen] = node.data; pathLen++; // it's a leaf, so print the path that led to here if (node.left==null && node.right==null) { printArray(path, pathLen); } else { // otherwise try both subtrees printPaths(node.left, path, pathLen); printPaths(node.right, path, pathLen); }}/** Utility that prints ints from an array on one line.*/private void printArray(int[] ints, int len) { int i; for (i=0; i<len; i++) { System.out.print(ints[i] + " "); } System.out.println();}

12. countTrees() Solution (Java)

/** For the key values 1...numKeys, how many structurally unique binary search trees are possible that store those keys? Strategy: consider that each value could be the root. Recursively find the size of the left and right subtrees.*/public static int countTrees(int numKeys) { if (numKeys <=1) { return(1); } else { // there will be one value at the root, with whatever remains // on the left and right each forming their own subtrees. // Iterate through all the values that could be the root... int sum = 0; int left, right, root; for (root=1; root<=numKeys; root++) { left = countTrees(root-1); right = countTrees(numKeys - root); // number of possible trees with this root == left*right sum += left*right; } return(sum); }}

13. isBST1() Solution (Java)

/** Tests if a tree meets the conditions to be a binary search tree (BST).*/public boolean isBST() { return(isBST(root));}/** Recursive helper -- checks if a tree is a BST using minValue() and maxValue() (not efficient).*/private boolean isBST(Node node) { if (node==null) return(true); // do the subtrees contain values that do not // agree with the node? if (node.left!=null && maxValue(node.left) > node.data) return(false); if (node.right!=null && minValue(node.right) <= node.data) return(false); // check that the subtrees themselves are ok return( isBST(node.left) && isBST(node.right) );}

14. isBST2() Solution (Java)

/** Tests if a tree meets the conditions to be a binary search tree (BST). Uses the efficient recursive helper.*/public boolean isBST2() { return( isBST2(root, Integer.MIN_VALUE, Integer.MAX_VALUE) );}/** Efficient BST helper -- Given a node, and min and max values, recurs down the tree to verify that it is a BST, and that all its nodes are within the min..max range. Works in O(n) time -- visits each node only once.*/private boolean isBST2(Node node, int min, int max) { if (node==null) { return(true); } else { // left should be in range min...node.data boolean leftOk = isBST2(node.left, min, node.data); // if the left is not ok, bail out if (!leftOk) return(false); // right should be in range node.data+1..max boolean rightOk = isBST2(node.right, node.data+1, max); return(rightOk); }}